It is generally known that the intervals of the equal tempered scale in popular use today are slightly out of tune in relation to pure harmony. Chords made from the intervals of this scale are disturbed by beats caused by this inexact tuning, resulting in dissonance. In contrast, tones derived from intervals of the just intonation scale form perfect harmonies, when sounded together. When a cappella choral singers sing or well trained chamber groups use unfretted instruments (violin, viola, cello), the pure harmonies of just intonation are heard. The equally tempered intervals were fixed in the seventeenth century to overcome mechanical difficulties in changing keys in fixed tone instruments like the piano, and fretted instruments like the guitar. In music dominated by the equally tempered intervals of the piano and guitar, pure harmonies are lost.
Just intonation intervals that create pure harmony can be defined by ratios of whole numbers such as 1:1, 2:1, 3:2, 4:3, and 5:4. Strings divided into these precise lengths give the same pure harmonies that singers had discovered naturally by ear. However, the tones created by these intervals are not entirely interchangeable when the key or chordal root of the music changes. That is, when the frequency of the tonic or key tone changes, a new musical scale is defined by the perfect ratios as applied to the new key tone. If the singers modulate the key from a key tone A(1:1) up to the key tone B(9:8 of key tone A) so as to define a new scale, some of the tones in the original scale will be found in the new scale, but not all; some tones of the new scale will be different. The D note played as a Fourth (4:3) of the key tone A is not the same frequency as the D note played as a Minor Third (6:5), of the key tone B. They are different because in the first case D is 4/3 the frequency of A, whereas in the second case D is 6/5 of 9/8 the frequency of A. These two values are different by a small ratio: 81:80. Modern music makes them equal by splitting the difference between both notes. This is only one example of the errors of the equal tempered scale.
Staying in perfect tune while changing keys is not difficult for singers or for players of instruments that allow any tone to be played, for example a violin. But fixed-tone instruments like the organ, clavichord, harpsichord and piano had to be altered or tempered in order to play in more than one key.
In the seventeenth century, the scale of "equal temperament," was developed fixing 12 equal intervals into an octave, thereby allowing all fixed tones to be used in every key. In 1685 German organist and music theorist Andreas Werckmeister, and Prussian musician Johann Neidhardt calculated the equal intervals as the 12th roots of the powers of two (2.sup.1, 2.sup.2, 2.sup.3, 2.sup.4, 2.sup.5, 2.sup.6, 2.sup.7, 2.sup.8, 2.sup.9, 2.sup.10, 2.sup.11, 2.sup.12,). This solved the problem of easy modulation for the pianos, but at the cost of throwing every interval out of pure tune.
Mechanical solutions to the problem of key modulation in just intonation were proposed by Hermann Helmholtz, Perronet Thompson, Henry Poole and others, but were simply too cumbersome and too limited to offer complete just intonation in all keys.
U.S. Pat. No. 3,821,460 to Motorola Inc. discloses an electronic keyboard capable of being tuned to equal temperament and just intonation, using programmable frequency dividers. The tuning, however, is not instantaneous, and the instrument can not be used for playing while allowing for modulation and chordal change in real time, but was rather meant as a static instructional tool. Furthermore, the keyboard does not realize true and complete just intonation scales.
U.S. Pat. No. 3,871,261 to Wells correctly pointed out that "the `equal tempered` system has virtually gained universal acceptance . . . but does not eliminate the beats" caused by notes "not perfectly in tune." His invention proposes 12 frequency modifiers (12 potentiometers) for each key, to render the pitch of each note adjustable, and a key selection device to switch musical keys. Wells' scales are not truly just in all cases, and the combination tones and overtones create disturbing beats. Furthermore, there is no provision for changing chordal root within a given key.
Electronic keyboard manufacturers began introducing various microtuning features in 1985 using logarithmic cents as a micro tuning unit. Keyboards and tone generators were produced with preset alternative scales including so-called "Pure" scales in as many as 12 major and 12 minor diatonic scales. To access one of these scales, the user has to step through many menu choices, and therefore modulating to another key during a composition is out of the question. Also, no provision is made for chordal root changes.
U.S. Pat. No. 4,152,964 to Waage discloses an electronic system to approximate just intonation by retaining "the tempered fourths and fifths," and shifting "the pitch of certain notes to correct the larger tuning errors of the scale." This invention was only an approximation of just intonation.
U.S. Pat. No. 4,248,119 to Yamada is a pitch correction gate system that attempts to detect chord structure and then alter tones from equal temperament to just intonation as chords are being played. This approach is impractical because the mixture of equal temperament and just intonation is more dissonant than tempered tuning alone.
U.S. Pat. No. 4,434,696 to Conviser recognized that "the influences of fixed-pitch instruments have contributed to a loss of correct pitch and have caused vocalists and instrumentalists not constrained by fixed pitch to sing and play `out of tune` either for equally tempered or `just` performance. Basic to this problem has been the lack of technological development in instruments for either tempered tuning or just intonation." The Conviser invention uses compound ratios to create the frequencies of equal temperament and just intonation. Conviser uses the correct just-intonation intervals from Ptolemy: 9/8, 5/4, 4/3, 3/2, 5/3, and 15/8, but derives the other intervals by multiplying "by 16/15 to obtain the flats . . . and by 25/24 to obtain the sharps." The resulting scale is not a correct nor a complete just intonation scale. No truly just scale is given, and there is no provision for the necessary tonal changes when changing chordal root within a given key.
U.S. Pat. No. 4,498,363 to Shimada disclosed a "just intonation electronic keyboard instrument". The keyboard comprised "a plurality of tonality selection switches for selecting each key from among twenty-four just intonation keys . . . " It noted that keyboard instruments which are tuned according to equal temperament are unfit for use in teaching during chorus practice. The patent describes 12 major diatonic scales, and twelve minor diatonic scales, but not complete chromatic scales. The invention is intended for choral practice, and there is no provision for changing the tuning in real time nor is there any provision for chordal root changes.
U.S. Pat. No. 4,796,509 to Yamaha Corporation of Japan disclosed an electronic tuning apparatus based on both equal temperament and just intonation scales. This apparatus generates a scale based on a reference signal, and displays a tone name for each frequency of the scale. The tuner can accommodate a single just intonation scale, but does not provide for chordal root changes as a composition is being played.
The Yamaha YMF262 FM Operator Type L3 chip was developed as a sound source for computer musical keyboards and tone generators. It is also used on many commercially available audio cards. This chip contains a frequency modulation sound source which may be controlled by software. All functions of the synthesizer are controlled by data written to its register array. The function for sending the frequency requires that the frequency be multiplied by 1.31072, rounded off to the next whole number, and then sent to a 10 bit address on the chip. This rounding-off makes it impossible to attain the simple fractions required for perfect just intonation harmonies.
U.S. Pat. No. 4,860,624 to Dinnan attempted to solve overtone collision, or dissonance. However, only some of the ratios given by Dinnan are correct just intonation intervals. Others have no relationship to historically used just scale intervals, and they create most unusual harmonies that cannot be considered Just or Pure. The Dinnan invention makes no provision for altering the scale when changing chordal root within a given key.
In view of the foregoing review of the prior art, and the failure of previous proposals to solve the problem of pure intonation for fixed-tone musical instruments, one of the objects of the present invention is to create a just intonation system that overcomes the aforementioned disadvantages and answers all the requirements of pure intonation including ease of play and modulation of both key and chordal root while playing.
The failure of the previously proposed solutions is that they are only half-measures at best, and do not offer a comprehensive just intonation system. To be practical for musicians a just intonation system must be comprehensive and perfect for all chordal roots, all keys, all inversions of chords, and in relation to all overtones and combination tones. It must also allow dynamic play in real time with instantaneous switching of key and root while playing the notes.